Quantifying total uncertainty in physicsinformed neural networks for solving forward and inverse stochastic problems
Abstract
Physicsinformed neural networks (PINNs) have recently emerged as an alternative way of numerically solving partial differential equations (PDEs) without the need of building elaborate grids, instead, using a straightforward implementation. In particular, in addition to the deep neural network (DNN) for the solution, an auxiliary DNN is considered that represents the residual of the PDE. The residual is then combined with the mismatch in the given data of the solution in order to formulate the loss function. This framework is effective but is lacking uncertainty quantification of the solution due to the inherent randomness in the data or due to the approximation limitations of the DNN architecture. Here, we propose a new method with the objective of endowing the DNN with uncertainty quantification for both sources of uncertainty, i.e., the parametric uncertainty and the approximation uncertainty. We first account for the parametric uncertainty when the parameter in the differential equation is represented as a stochastic process. Multiple DNNs are designed to learn the modal functions of the arbitrary polynomial chaos (aPC) expansion of its solution by using stochastic data from sparse sensors. We can then make predictions from new sensor measurements very efficiently with the trained DNNs. Moreover, we employ dropout to quantify the uncertainty of DNNs in approximating the modal functions. We then design an active learning strategy based on the dropout uncertainty to place new sensors in the domain in order to improve the predictions of DNNs. Several numerical tests are conducted for both the forward and the inverse problems to demonstrate the effectiveness of PINNs combined with uncertainty quantification. This NNaPC new paradigm of physicsinformed deep learning with uncertainty quantification can be readily applied to other types of stochastic PDEs in multidimensions.
 Publication:

Journal of Computational Physics
 Pub Date:
 November 2019
 DOI:
 10.1016/j.jcp.2019.07.048
 arXiv:
 arXiv:1809.08327
 Bibcode:
 2019JCoPh.39708850Z
 Keywords:

 Physicsinformed neural networks;
 Uncertainty quantification;
 Stochastic differential equations;
 Arbitrary polynomial chaos;
 Dropout;
 Mathematics  Analysis of PDEs;
 Physics  Computational Physics;
 Statistics  Machine Learning
 EPrint:
 doi:10.1016/j.jcp.2019.07.048